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with dimension L2/T and u- 
nit St (abbreviation of Stokes) = 1 cmz/s in the CGS system or m2/s( = l o 4 St) in 
the SI system. 
h is a parameter related to the inflation of volume. As stated before, the Stokes 
hypothesis states that the trace of the tensor z equals zero, so from Eq. (1. 2. 39), h 
and ,u must satisfy the condition 3h+2,u=O. The quantity I+2,u/3 is called the sec- 
ond viscosity, while p is the first viscosity. For a Stokes' fluid, the second viscosity 
is equal to zero, A= - 2p/3 (cf. Eq. ( 1. 2. 4 1 ) ) . Stokes' hypothesis is accurate e- 
nough in engineering applications, at least for gases and Newtonian fluids. 
For a Stokes' fluid, it can be proved that only the viscosity normal stress exists 
when inflation rate in the same direction is not equal to the mean value over those in 
the three coordinate directions, while viscosity shear stress is proportional to shear de- 
formation rate. Especially in the case of liquids, there are two sources of viscosity 
shear stress. One is the momentum exchange and viscosity dissipation stemming from 
random thermal motions of molecules, but its magnitude is relatively small. The oth- 
er is the continuous redistribution of molecules due to the velocity gradient, incurring 
a rotation of the net force field among contiguous molecules. 
k-,uC,,(or ,uCJ , D ~ P 
1111. NAVIER-STOKES ( N S ) EQUATIONS 
1. NS equation in vector form 
Introducing the constitutive equations for isotropic Newtonian Stokes fluid (i. 
e. , under the assumptions that ,u=const, 3I+2,u=0) , Eqs. (1. 2. 28) and (1. 2. 
4 1 ) , into the equation of motion, Eq. (1. 2. 25 ) , we obtain the NS equation 
where V ( V V ) is a divergence of the gradient of vector V . In a rectangular coor- 
dinate system, the operator is called the Laplace operator (Laplacian) , also denoted 
by V *(or A ) , since it can be expanded by taking a formal operation like scalar prod- 
uct 
14 
(1. 2. 45) 
In this case, the components of V ( V V ) can be obtained by applying V to the 
correspondingcomponentsof V , i .e. , V (V V ) = V 2 V = V VV (SOV V is 
called a quasi-vector) . However, this relation does not hold in a more general coordi- 
nate system, when the operator should be expanded into 
(1. 2. 46) 
For an incompressible fluid, the last term on the right-hand side of Eq. (1. 2. 44) e- 
quals zero. If we have also an ideal fluid ( p = 0) , a reduced form of the NS equa- 
tions is called Euler equations. But even if p# 0, when all the derivatives &,/ax, are 
sufficiently small, the flow can still be approximated by the Euler equations. When 
DV/Dt= 0, the NS equations are reduced to a linear system, Stokes equations. The 
associated flow, Stokes flow (not to be confused with the Stokes fluid) , occurs when 
the inertial force is much smaller than the viscous force. 
We note in passing that historically only the equations of motion were called the 
NS equation. Now this term often means the complete system, including the equation 
of continuity (and additionally, the equation of energy for a general compressible 
fluid). 
There are important differences in structure and solution between the NS equa- 
tions for compressible and incompressible fluids. For incompressible fluids, only ve- 
locity appears in the equation of continuity, containing no density and pressure. This 
equation, as it is not in a complete form, is only partly coupled with the equation of 
motion and thus can be considered as a constraint on the velocity field. Pressure is in 
an unequal position to velocity, and cannot be determined by some thermodynamic 
condition (e. g. , equation of state). In numerical solutions, pressure and velocity 
are often obtained alternately in an iterative process. A difficulty lies in that the ve- 
locity obtained from the equation of motion under a given pressure generally cannot 
fulfill the equation of continuity. 
As for a compressible fluid, density appears in the equation of continuity, which 
is fully coupled with the equation of motion. If we take a fixed region in a flow, the 
pressure gradient at its boundary determines a velocity of the fluid leaving that re- 
gion, based on the equation of motion. That velocity incurs conversely a variation of 
density in the region based on the equation of continuity, and this has a feedback ef- 
fect on the pressure through the equation of state. In numerical solutions, pressure 
and velocity are often solved for simultaneously, and the 'elastic' constraint provided 
by the equation of continuity make the procedure easier due to inter-adjustability be- 
tween pressure and velocity. 
2. NS equations in rectangular coordinate system for incompressible flow 
Later in this section we discuss incompressible flow only, but thereafter we shall 
study 2-D shallow-water flow from the viewpoint of compressible flow. The equation 
of continuity is 
(1. 2. 47) 
15 
Only the equation of motion in the x1 direction is listed below 
For the sake of brevity, it may be expressed by the use of a Cartesian tensor 
Eqs. (1. 2. 47) and (1. 2. 48) may be written concisely as 
(1. 2. 49) 
(1. 2. 5 0 ) 
(1. 2. 51) 
(1. 2. 52) 
We note in passing that space coordinates and velocity components will occasion- 
ally be denoted by (xl , x 2 , x3) and (u l , u 2 , us> , and sometimes by ( x , y , z ) and 
( u , v , w). The latter are chiefly used in the 2-D case. 
3. The NS equations expressed in terms of the stream function for 2-D incompre 
sible inviscid irrotational flows 
In the 2-D cases (plane or axis-symmetry) , it is possible to eliminate the pres- 
sure p from Eq. (1. 2. 52). Moreover, since for incompressible flows, p does not ap- 
pear in the equation of continuity, u and v can be combined into one unknown func- 
tion, so that only one equation is left. 
Defining the stream function @ by 
(1.2. 53) 
and introducing the above equation into the NS equations with p= 0 , we obtain 
(1. 2 .54) 
(1.2. 55) 
Taking derivatives of the above two equations with respect to y and x respective- 
ly , and then eliminating p , we have 
(1. 2. 56) 
When the flow is vortex-free, i. e. , the vorticity equals zero everywhere, the 
condition V X V = 0 can be expanded in the 2-D case, yielding 
16 
= o a u a v --- 
ag ax 
(1.2. 57) 
Introducing Eq. (1. 2. 57) into Eq. (1. 2. 53) , we get a Laplace equation satis- 
fied by the stream function 
v2*= 0 (1. 2. 58) 
Such an irrotational flow is called a potential flow. 
Substitute Eq. (1. 2. 58) into Eq. (1. 2. 561, and solve for the stream function 
under a given boundary condition. Then by taking a derivative in Eq. (1. 2. 53) we 
get a solenoidal velocity field also satisfying the NS equations. Lastly, substitute the 
velocity into the NS equations to obtain the pressure gradient. The whole problem has 
now been solved. An advantage of this approach is that the difficulty stemming from 
nonlinear convective terms can be avoided. 
In an extension to the 3-D case, a scalar velocity potential function can be in- 
troduced to replace the three velocity components. For a steady, irrotational and 
isentropic flow with pressure as the only external force, the NS equations can again 
be reduced to a single PDE in terms of the velocity potential. 
4. Simplification of the equation of motion for incompressible flows 
For a general incompressible flow, the stream function cannot be used advan- 
tangeously, so it is necessary to adopt another approach in order to cancel out the e- 
quation of continuity and so eliminate pressure from the equation of motion. 
Firstly, we introduce a theorem : Any vector field w defined on a region can be 
uniquely decomposed into w=V+ V p , where vector field V satisfies: (i) div V= 0; 
(ii) V n= 0 , where n is an outward normal vector to the boundary of Sa. Physical- 
ly, it expresses the continuity requirement and the boundary condition that vector Ir is 
in its tangential direction. 
Based on the theorem, we define a linear